Problem 5 - The Calculus with Analytic Geometry

Question

Find the derivative of the given function. $$ f(x)=(x+4)^{-2} $$

Step-by-Step Solution

Answer

\[ \frac{d}{dx} (x + 4)^{-2} = \frac{-2}{(x + 4)^3} \]

Step 1: Identify the Function

The given function is: \[ f(x) = (x+4)^{-2} \]

Step 2: Apply the Power Rule

Use the power rule for differentiation, which states that if \( f(x) = x^n \), then \( f'(x) = n x^{n-1} \). Here, let \( u = x + 4 \). Thus, \( f(u) = u^{-2} \).

Step 3: Differentiate with respect to u

Apply the power rule to \( f(u) \): \[ \frac{d}{du} (u^{-2}) = -2 u^{-3} = \frac{-2}{u^3} \]

Step 4: Differentiate u with respect to x

Since \( u = x + 4 \), the derivative of \( u \) with respect to \( x \) is: \[ \frac{du}{dx} = 1 \]

Step 5: Apply the Chain Rule

The chain rule states that if \( y = f(u) \) and \( u = g(x) \), then \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). Here, \( \frac{d}{dx} (x+4)^{-2} = \frac{d}{du} (u^{-2}) \cdot \frac{du}{dx} = \frac{-2}{(x+4)^3} \cdot 1 \).

Step 6: Simplify the Result

Combine the results: \[ \frac{d}{dx} (x + 4)^{-2} = \frac{-2}{(x + 4)^3} \]

Key Concepts

Power Rule

When differentiating functions with exponents, the power rule is a key technique. This rule states that if you have a function such as \( f(x) = x^n \), then its derivative, or rate of change, is \( f'(x) = n x^{n-1} \). This means you bring the exponent down as a coefficient and then subtract one from the exponent.
Let's see an example: Suppose we need to find the derivative of \( f(x) = x^3 \). According to the power rule:

• Bring down the 3 (the exponent) as a coefficient.
• Subtract one from the exponent.

This gives us \( f'(x) = 3x^{3-1} = 3x^2 \).
In our problem, where we have \( f(u) = u^{-2} \), applying the power rule gives us \( f'(u) = -2 u^{-3} \). This technique is particularly useful for polynomials and functions that can be expressed with exponents.

Chain Rule

The chain rule is a powerful method in differentiation when dealing with composite functions. It tells us how to differentiate a function of a function. If you have two functions, say \( y = f(u) \) and \( u = g(x) \), then the chain rule states:

• \( y \, \ = f(g(x)) \)
• \( \frac{dy}{dx} \, \ = \frac{dy}{du} \, \cdot \frac{du}{dx} \)

The chain rule helps in breaking down the derivative process into simpler parts.
For the given function \( f(x) = (x+4)^{-2} \), let \( u = x+4 \). Then \( f(u) = u^{-2} \).
Applying the chain rule:

• First, find \( \frac{d}{du}(u^{-2}) \): Applying the power rule, we get \( \frac{-2}{u^3} \)
• Next, differentiate \( u = x+4 \) with respect to \( x \): We have \( \frac{du}{dx} = 1 \)
• Finally, multiply these results: \( \frac{d}{dx}(x+4)^{-2} = \frac{d}{du}(u^{-2}) \cdot \frac{du}{dx} = \frac{-2}{(x+4)^3} \cdot 1 = \frac{-2}{(x+4)^3} \)

Using the chain rule simplifies the task of differentiating composite functions.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate of change of a function. The derivative of a function gives a measure of how the function's output changes as its input changes.
For example, if you have the function \( f(x) \), its derivative \( f'(x) \) tells you how the function value changes for a small change in \( x \).
To differentiate a function:

• Identify the function and its form.
• Apply appropriate differentiation rules such as the power rule, product rule, quotient rule, or chain rule.
• Simplify the resulting expression.

In our problem, the function is \( f(x) = (x+4)^{-2} \). We used the power rule and chain rule to find its derivative, which is: \( \frac{d}{dx}(x+4)^{-2} = \frac{-2}{(x+4)^3} \).
Understanding differentiation is crucial for solving problems in physics, engineering, economics, and many other fields where rates of change are important.